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I've solved this puzzler. Step by step. First I've rotated the camera coordinate system so that it matches the camera in 0/0 orientation (pan/tilt) according to the mentioned requirements. I've created a fixed quaternion for that rotation. Then I've empirically figured out what axis I had to rotate by which angle. One after the other I've multiplied them with the current quaternion.

Now I have to correct orientation of the tf transform of the camera. Tough one.